Use Transformations To Graph The Function Defined By $h(x)=\frac{1}{2}|x-4|-5$. Identify The Parent Function. Then, List The Transformations Applied To The Parent Function Needed To Obtain The Graph Of $h$ In The Appropriate

Mar 8, 2025 by ADMIN 229 views

**Graphing Transformations of Functions**

Introduction

Graphing functions can be a complex task, especially when dealing with transformations. In this article, we will explore how to graph the function defined by $h(x)=\frac{1}{2}|x-4|-5$, and identify the parent function. We will also list the transformations applied to the parent function to obtain the graph of $h$.

Understanding the Parent Function

The parent function is the basic function that has not been transformed. In this case, the parent function is the absolute value function, which is defined as $f(x)=|x|$. This function has a V-shaped graph with its vertex at the origin (0, 0).

Graphing the Parent Function

To graph the parent function, we can use the following steps:

Plot the vertex at (0, 0).
Plot two points on either side of the vertex, one at (1, 1) and the other at (-1, 1).
Draw a smooth curve through the points, making sure that the curve is V-shaped.

Transforming the Parent Function

Now that we have graphed the parent function, we can apply transformations to obtain the graph of $h(x)=\frac{1}{2}|x-4|-5$. The transformations applied to the parent function are:

Horizontal Shift: The graph of $h$ is shifted 4 units to the right. This means that the vertex of the graph is now at (4, -5).
Horizontal Compression: The graph of $h$ is compressed horizontally by a factor of 2. This means that the width of the graph is now half of the original width.
Vertical Shift: The graph of $h$ is shifted 5 units down. This means that the vertex of the graph is now at (4, -10).
Vertical Stretch: The graph of $h$ is stretched vertically by a factor of 1/2. This means that the height of the graph is now half of the original height.

Graphing the Function h(x)

To graph the function $h(x)=\frac{1}{2}|x-4|-5$, we can apply the transformations listed above to the parent function. The graph of $h$ is a V-shaped graph with its vertex at (4, -10). The graph is compressed horizontally by a factor of 2, and stretched vertically by a factor of 1/2.

Conclusion

In this article, we have graphed the function defined by $h(x)=\frac{1}{2}|x-4|-5$, and identified the parent function. We have also listed the transformations applied to the parent function to obtain the graph of $h$. The transformations applied to the parent function are a horizontal shift, horizontal compression, vertical shift, and vertical stretch.

Key Takeaways

The parent function is the absolute value function, which is defined as $f(x)=|x|$.
The graph of the parent function is a V-shaped graph with its vertex at the origin (0, 0).
The transformations applied to the parent function are a horizontal shift, horizontal compression, vertical shift, and vertical stretch.
The graph of $h(x)=\frac{1}{2}|x-4|-5$ is a V-shaped graph with its vertex at (4, -10).

Final Thoughts

Introduction

In our previous article, we explored how to graph the function defined by $h(x)=\frac{1}{2}|x-4|-5$, and identified the parent function. We also listed the transformations applied to the parent function to obtain the graph of $h$. In this article, we will answer some frequently asked questions about graphing transformations of functions.

Q&A

Q: What is the parent function of the function $h(x)=\frac{1}{2}|x-4|-5$?

A: The parent function of the function $h(x)=\frac{1}{2}|x-4|-5$ is the absolute value function, which is defined as $f(x)=|x|$. This function has a V-shaped graph with its vertex at the origin (0, 0).

Q: What are the transformations applied to the parent function to obtain the graph of $h(x)=\frac{1}{2}|x-4|-5$?

A: The transformations applied to the parent function to obtain the graph of $h(x)=\frac{1}{2}|x-4|-5$ are:

Horizontal Shift: The graph of $h$ is shifted 4 units to the right.
Horizontal Compression: The graph of $h$ is compressed horizontally by a factor of 2.
Vertical Shift: The graph of $h$ is shifted 5 units down.
Vertical Stretch: The graph of $h$ is stretched vertically by a factor of 1/2.

Q: How do I graph the function $h(x)=\frac{1}{2}|x-4|-5$?

A: To graph the function $h(x)=\frac{1}{2}|x-4|-5$, you can follow these steps:

Plot the vertex at (4, -10).
Plot two points on either side of the vertex, one at (5, -5) and the other at (3, -5).
Draw a smooth curve through the points, making sure that the curve is V-shaped.

Q: What is the vertex of the graph of $h(x)=\frac{1}{2}|x-4|-5$?

A: The vertex of the graph of $h(x)=\frac{1}{2}|x-4|-5$ is at (4, -10).

Q: How do I determine the transformations applied to the parent function?

A: To determine the transformations applied to the parent function, you can look at the function and identify the following:

Horizontal Shift: If the function is of the form $f(x-a)$, then the graph is shifted $a$ units to the right.
Horizontal Compression: If the function is of the form $f(ax)$, then the graph is compressed horizontally by a factor of $a$.
Vertical Shift: If the function is of the form $f(x)+b$, then the graph is shifted $b$ units up.
Vertical Stretch: If the function is of the form $af(x)$, then the graph is stretched vertically by a factor of $a$.

Q: Can I use the graphing calculator to graph the function $h(x)=\frac{1}{2}|x-4|-5$?

A: Yes, you can use the graphing calculator to graph the function $h(x)=\frac{1}{2}|x-4|-5$. Simply enter the function into the calculator and press the graph button.

Conclusion

In this article, we have answered some frequently asked questions about graphing transformations of functions. We have discussed the parent function, the transformations applied to the parent function, and how to graph the function $h(x)=\frac{1}{2}|x-4|-5$. We have also provided some tips on how to determine the transformations applied to the parent function and how to use the graphing calculator to graph the function.

Key Takeaways

The parent function of the function $h(x)=\frac{1}{2}|x-4|-5$ is the absolute value function, which is defined as $f(x)=|x|$. This function has a V-shaped graph with its vertex at the origin (0, 0).
The transformations applied to the parent function to obtain the graph of $h(x)=\frac{1}{2}|x-4|-5$ are a horizontal shift, horizontal compression, vertical shift, and vertical stretch.
The vertex of the graph of $h(x)=\frac{1}{2}|x-4|-5$ is at (4, -10).
You can use the graphing calculator to graph the function $h(x)=\frac{1}{2}|x-4|-5$.

Final Thoughts

Graphing transformations of functions can be a complex task, but by understanding the parent function and the transformations applied to it, we can easily graph the function. In this article, we have answered some frequently asked questions about graphing transformations of functions and provided some tips on how to determine the transformations applied to the parent function and how to use the graphing calculator to graph the function.